Wasserstein, Constant Speed Geodesic, Optimal Transport Map. Ambrosio Lemma 7.2.1. Disintegration.

Question

I'm reading Lemma 7.2.1 of Ambrosio Gradient Flows in Metric Spaces 2nd edition.

On the top of page 159 we have

\begin{align*} W_p(\mu_0,\mu_1)\leq& \Big( \int_{X\times X} \|x_1-x_3\|^p d\mu(x_1,x_3) \Big)^{1/p} \\ \leq& ...? \\ =&\Big(\int_{X\times X} \|x_1-x_2\|^p d\gamma(x_1,x_2) \Big)^{1/p}+\Big(\int_{X\times X} \|x_2-x_3\|^p d\eta(x_2,x_3) \Big)^{1/p} \\ =& W_p(\mu_0,\mu_t)+W_p(\mu_t,\mu_1) \\ =& tW_p(\mu_0,\mu_1)+(1-t)W_p(\mu_0,\mu_1)=W_p(\mu_0,\mu_1). \end{align*}

But I cant fill in the blanks in the line of argument, to be honest I don't understand the notation $\|x_1-x_3\|_{L^p(\mu,X)}$ I would read this as the Lp norm of function $x_1$ but obviously it doesn't mean this. Can someone explain the reasoning?

Answer 1

This is a shortcut for $\|f_1-f_3\|_{L^p(\mu,X)}$ with $f_1(x_1,x_3)=x_1$ and $f_3(x_1,x_3)=x_3$.

Answer 2

If you mean the "$\le\dots$" part, that's because of the triangle inequality of the norm.